Chapter 12: Static Equilibrium Thursday March 19th

Reading: up to page 195 in Ch. 12

• Review of Torque and Newton’s 2nd law • The requirements for equilibrium • Static equilibrium • Stable and unstable equilibrium • Examples, demonstrations and iclicker

• I will be away on Tuesday – Dr. Hori to cover for me. • My office hours today from 11-12:30; also next Thu. 11-noon. • Tuesday’s lecture jumps back to Chapter 8 on Gravity. • Mini-Exam next Thursday 26th (LONCAPA #13-17).

Review: Torque and Newton’s 2nd Law

τ = !r ×

!F = rF sinθDefinition:

Newton’s 2nd law: τ = IαRotational equivalent of force

Rotational acceleration

Rotational equivalent of mass

Equilibrium A system of objects is said to be in equilibrium if:

1.  The linear momentum P of its center of mass is constant. 2.  Its angular momentum L about its center of mass, or about

any other point, is also constant.

If, in addition, L and P are zero, the system is said to be in static equilibrium.

Examples of dynamic equilibrium: • Ice hockey puck sliding on frictionless ice • An object in free fall having reached terminal velocity • A ball rolling unimpeded on a horizontal surface • A bicycle traveling at constant velocity

Examples of static equilibrium: • A ladder leaning against a wall • A static pile of rocks, sand or grain • A house of cards • A gymnast performing the crucifix

Equilibrium A system of objects is said to be in equilibrium if:

1.  The linear momentum P of its center of mass is constant. 2.  Its angular momentum L about its center of mass, or about

any other point, is also constant.

If, in addition, L and P are zero, the system is said to be in static equilibrium.

Examples of non-equilibrium situations: • An object falling in a vacuum under gravity • A rocket during launch • Most of the time you are on a roller coaster ride • A ladder leaning against a wall if the contacts between the wall and the ground are frictionless

Equilibrium A system of objects is said to be in equilibrium if:

1.  The linear momentum P of its center of mass is constant. 2.  Its angular momentum L about its center of mass, or about

any other point, is also constant.

If, in addition, L and P are zero, the system is said to be in static equilibrium.

The requirements of equilibrium (1) Translational motion of a body is governed by Newton's 2nd law:

!Fnet =

d!P

dt⇒!Fnet = 0

(2) Rotational motion of a body is governed by Newton's 2nd law in its angular momentum form:

!τ net =

d!L

dt⇒!τ net =

!ri ×!Fi( )

aboutany point

∑ = 0

1.  The vector sum of all the external forces that act on a body must be zero.

2.  The vector sum of all the external torques that act on a body, measured about any axis, must also be zero.

*Turns out that if this is true for one point, it is true for any point.

*

The requirements of equilibrium 1.  The vector sum of all the external forces that act on a

body must be zero. 2.  The vector sum of all the external torques that act on

a body, measured about any axis, must also be zero.

!Fnet ,x = 0

!τ net ,x = 0

!Fnet ,y = 0

!τ net ,y = 0

!Fnet ,z = 0

!τ net ,z = 0

Balance of torques

Balance of forces

3.  Linear & angular momenta of the system must be zero.

One more requirement for static equilibrium:

Stable versus unstable equilibrium:

M

Stable Unstable

M

A system of objects is said to be in equilibrium if: 1.  The linear momentum P of its center of mass is constant. 2.  Its angular momentum L about its center of mass, or about

any other point, is also constant.

If, in addition, L and P are zero, the system is said to be in static equilibrium.

Stable and Unstable Equilibrium

Stable and Unstable Equilibrium A system of objects is said to be in equilibrium if:

1.  The linear momentum P of its center of mass is constant. 2.  Its angular momentum L about its center of mass, or about

any other point, is also constant.

If, in addition, L and P are zero, the system is said to be in static equilibrium.

: d 2Udx2 < 0

: d 2Udx2 > 0

Mg

N Mg=

Stable

Mg

N Mg=

Unstable

!Fnet =

dUdx

= 0∑

x

Stable and Unstable Equilibrium A system of objects is said to be in equilibrium if:

1.  The linear momentum P of its center of mass is constant. 2.  Its angular momentum L about its center of mass, or about

any other point, is also constant.

If, in addition, L and P are zero, the system is said to be in static equilibrium.

Metastable equilibrium

Stable equilibrium

Examples: • An avalanche • Heat pack • A super-cooled liquid

Stable and Unstable Equilibrium A system of objects is said to be in equilibrium if:

1.  The linear momentum P of its center of mass is constant. 2.  Its angular momentum L about its center of mass, or about

any other point, is also constant.

If, in addition, L and P are zero, the system is said to be in static equilibrium.

x y

dUdx

= dUdy

= 0

d 2Udx2 > 0

d 2Udy2 < 0

Unstable